Question Type 3: Related rates with geometric constraints Exercises

A balloon is rising vertically from a launch point at a constant speed of $4 \text{ m s}^{-1}$. A cyclist rides along a straight horizontal road that passes directly under the launch point. The cyclist is moving away from the launch point at a constant speed of $12 \text{ m s}^{-1}$.

Find the rate at which the distance between the cyclist and the balloon is changing when the cyclist is $50 \text{ m}$ from the launch point and the balloon is at a height of $80 \text{ m}$.

An inverted conical tank (height $10 \text{ m}$, base radius $3 \text{ m}$) is leaking water at $2 \text{ m}^3/\text{min}$. How fast is the water depth decreasing when the water is $5 \text{ m}$ deep?

A plane flying horizontally at $200 \text{ m s}^{-1}$ passes directly over a radar station at altitude $3 \text{ km}$. How fast is the distance from the plane to the station increasing $30 \text{ s}$ after it passes over?

In a right-angled triangle, one leg is fixed at $10 \text{ m}$ while the other leg is increasing at a rate of $0.4 \text{ m s}^{-1}$.

Find the rate at which the hypotenuse is changing when the growing leg is $6 \text{ m}$.

A spherical snowball melts so that its radius decreases at $0.2 \text{ cm/min}$. At what rate is its volume changing when the radius is $15 \text{ cm}$?

A square has side length increasing at $0.5 \text{ cm/s}$. How fast is its diagonal increasing when the side length is $8 \text{ cm}$?

The area of a circle is increasing at $20 \text{ cm}^2\text{ s}^{-1}$. How fast is its radius increasing when the radius is $4 \text{ cm}$?

A $10 \text{ m}$ ladder leans against a wall so that the angle $\theta$ with the ground changes as the base slides outward at $0.8 \text{ m s}^{-1}$. How fast is $\theta$ changing when the base is $6 \text{ m}$ from the wall?

Water is being poured into a conical tank of height 12 m and base radius 5 m. The water level rises at 0.3 m/min. How fast is the volume of water increasing when the depth is 4 m?

Water flows into a conical vessel at $5 \text{ L/min}$. The vessel has height $4 \text{ m}$ and base radius $2 \text{ m}$. How fast is the water level rising when the depth is $2 \text{ m}$? ($1 \text{ L}=0.001 \text{ m}^3$)

The radius of a right circular cylinder of fixed height $10 \text{ cm}$ is increasing at $0.1 \text{ cm s}^{-1}$. How fast is the volume increasing when the radius is $5 \text{ cm}$?

A ladder $10\,\text{m}$ long leans against a vertical wall. The bottom is pulled away from the wall at $2\,\text{m/s}$. How fast is the top sliding down the wall when the bottom is $6\,\text{m}$ from the wall?